p Orbital flat band and Dirac cone in the electronic honeycomb lattice
Thomas S. Gardenier, Jette J. van den Broeke, Jesper R. Moes, Ingmar Swart, Christophe Delerue, Marlou R. Slot, Cristiane Morais Smith, Daniel Vanmaekelbergh
Theory anticipates that the in-plane px, py orbitals in a honeycomb lattice lead to new and potentially useful quantum electronic phases. So far, p orbital bands were only realized for cold atoms in optical lattices and for light and exciton-polaritons in photonic crystals. For electrons, in-plane p orbital physics is difficult to access since natural electronic honeycomb lattices, such as graphene and silicene, show strong s-p hybridization. Here, we report on electronic honeycomb lattices prepared on a Cu(111) surface in a scanning tunneling microscope that, by design, show (nearly) pure orbital bands, including the p orbital flat band and Dirac cone.
The zigzag edge graphene nanoribbon, which is an antiferromagnetic insulator, is found from the density-function theory calculation to display a robust Dirac point after N and B doping at the zigzag edge. More interestingly, we found that the robust Dirac point is a common feature of the honeycomb structure nanoribbon with appropriate edge sites. The zigzag edge honeycomb nanoribbon is therefore expected to provide a very useful platform for material design and development.
We derive the low-energy Hamiltonian for a honeycomb lattice with anisotropy in the hopping parameters. Taking the reported Dirac Hamiltonian for the anisotropic honeycomb lattice, we obtain its optical conductivity tensor and its transmittance for normal incidence of linearly polarized light. Also, we characterize its dichroic character due to the anisotropic optical absorption. As an application of our general findings, which reproduce the case of uniformly strained graphene, we study the optical properties of graphene under a nonmechanical distortion.
We study the band structure of electrons hopping on a honeycomb lattice with 1/q (q integer) flux quanta through each elementary hexagon. In the nearest neighbor hopping model the two bands that eventually form the n = 0 Landau level have 2q zero energy Dirac touchings. In this work, we study the conditions needed for these Dirac points and their stability to various perturbations. We prove that these touchings and their locations are guaranteed by a combination of an anti-unitary particle-hole symmetry and the lattice symmetries of the honeycomb structure. We also study the stability of the D…
We study the electronic structure and the phase diagram of non-interacting fermions confined to hexagonal optical lattices. In the first part, we compare the properties of Dirac points arising in the eigenspectrum of either honeycomb or triangular lattices. Numerical results are complemented by analytical equations for weak and strong confinements. In the second part we discuss the phase diagram and the evolution of Dirac points in honeycomb lattices applying a tight-binding description with arbitrary nearest-neighbor hoppings. With increasing asymmetry between the hoppings the Dirac points ap…
We derive the boundary condition for the Dirac equation corresponding to a tight-binding model on a two-dimensional honeycomb lattice terminated along an arbitary direction. Zigzag boundary conditions result generically once the boundary is not parallel to the bonds. Since a honeycomb strip with zigzag edges is gapless, this implies that confinement by lattice termination does not in general produce an insulating nanoribbon. We consider the opening of a gap in a graphene nanoribbon by a staggered potential at the edge and derive the corresponding boundary condition for the Dirac equation. We a…
Silicene, a sheet of silicon atoms in a honeycomb lattice, was proposed to be a new Dirac-type electron system similar as graphene. We performed scanning tunneling microscopy and spectroscopy studies on the atomic and electronic properties of silicene on Ag(111). An unexpected $\sqrt{3}\times \sqrt{3}$ reconstruction was found, which is explained by an extra-buckling model. Pronounced quasi-particle interferences (QPI) patterns, originating from both the intervalley and intravalley scattering, were observed. From the QPI patterns we derived a linear energy-momentum dispersion and a large Fermi…
Carbon allotropes such as diamond, nano-tube, Fullerene, and Graphene, have unique lattice symmetries of crystal lattice, but these are topologically trivial. We have proposed a topologically-nontrivial allotrope, named Hopfene, which has three-dimensional (3D) arrays of Hopf-links to bind 2D Graphene sheets both vertically and horizontally. Here, we describe the electronic structures of Hopfene by simple tight-binding calculations. We confirmed the original Dirac points of 2D Graphene were topologically protected upon the introduction of the Hopf links, and low-energy excitations are describe…
We study theoretically two-dimensional single-crystalline sheets of semiconductors that form a honeycomb lattice with a period below 10 nm. These systems could combine the usual semiconductor properties with Dirac bands. Using atomistic tight-binding calculations, we show that both the atomic lattice and the overall geometry influence the band structure, revealing materials with unusual electronic properties. In rocksalt Pb chalcogenides, the expected Dirac-type features are clouded by a complex band structure. However, in the case of zinc-blende Cd-chalcogenide semiconductors, the honeycomb n…
We consider different generalizations of the honeycomb lattice to three dimensional structures. We address the family of the hyper-honeycomb lattice, which is made up of alternating layers of 2D honeycomb nano-ribbons, with each layer rotated by $π/2$ with the layer below. When the orbitals of the lattice sites are symmetric with respect to the planes of the trigonal links, these structures can produce a Dirac loop, a closed line of Dirac nodes in momentum space. For orbitals that break that symmetry, such as the carbon $p$-wave orbitals, hyper-honeycomb lattices do not possess the loop struct…
Powered by effective parquet nesting a commensurate chiral honeycomb superstructure in a trigonally packed transition metal dichalcogenide TaSe$_2$ results in a Dirac cone anomaly in one-particle excitation spectrum. However, the formation of the well defined Dirac point seems to be hindered by effective scattering on 2D plasmons.
A honeycomb lattice system has four types of Dirac electrons corresponding to the spin and valley degrees of freedom. We consider a state that contains only one type of massless electrons and three types of massive ones, which we call the single Dirac-cone state. We analyze quantum Hall (QH) effects in this state. We make a detailed investigation of the Chern and spin-Chern numbers. We make clear the origin of unconventional QH effects discovered in graphene. We also show that the single Dirac-cone state may have arbitrary large spin-Chern numbers in magnetic field. Such a state will be genera…
We show by means of ab initio calculations and tight-binding modeling that an oxide system based on a honeycomb lattice can sustain topologically non-trivial states if a single orbital dominates the spectrum close to the Fermi level. In such situation, the low energy spectra is described by two Dirac equations that become non-trivially gapped when spin-orbit coupling (SOC) is switched on. We provide one specific example for this but the recipe is general. We discuss a realization of this starting from a conventional spin-a-half honeycomb antiferromagnet whose states close to the Fermi energy a…
We investigate the spectrum and the dispersion relation of the Schrödinger operator with point scatterers on a triangular lattice and a honeycomb lattice. We prove that the low-level dispersion bands have conic singularities near Dirac points, which are the vertices of the first Brillouin Zone. The existence of such conic dispersion bands plays an important role in various electronic properties of honeycomb-structured materials such as graphene. We then prove that for a honeycomb lattice, the spectra generated by higher-level dispersion relations are all connected so the complete spectrum cons…
In addition to the known pseudomagnetic field, nonuniform strains independently induce a position-dependent Fermi velocity (PDFV) in graphene. Here we demonstrate that, due to the presence of a PDFV, the Dirac fermions on a nonuniform (strained) honeycomb lattice may experiment a sort of magnetic effect, which is linearly proportional to the momentum of the quasiparticle. As a consequence, the quasiparticles have a sublinear dispersion relation. Moreover, we analyze the general consequence of a PDFV on the Klein tunneling of electrons through pseudomagnetic barriers. In particular, we report a…
Two-dimensional lattices of coupled micropillars etched in a planar semiconductor microcavity offer a workbench to engineer the band structure of polaritons. We report experimental studies of honeycomb lattices where the polariton low-energy dispersion is analogous to that of electrons in graphene. Using energy-resolved photoluminescence we directly observe Dirac cones, around which the dynamics of polaritons is described by the Dirac equation for massless particles. At higher energies, we observe p orbital bands, one of them with the nondispersive character of a flatband. The realization of t…
Motivated by the recent proposal of Bosonic Dirac materials (BDM), we revisited the Ising model on a honeycomb lattice in the presence of the longitudinal and transverse fields. We apply linear spin-wave theory to obtain the magnon dispersion and its degenerated points. These special degenerated points emerge on the excitation spectrum as a function of the external fields and can be identified as Bosonic Dirac Points (BDP). Since that, in the vicinity of these points the Magnons becomes massless with a linear energy spectrum as well as insensible in relation to weak impurity, exactly as it occ…
Compression dramatically changes the transport and localization properties of graphene. This is intimately related to the change of symmetry of the Dirac cone when the particle hopping is different along different directions of the lattice. In particular, for a critical compression, a semi-Dirac cone is formed with massless and massive dispersions along perpendicular directions. Here we show direct evidence of the highly anisotropic transport of polaritons in a honeycomb lattice of coupled micropillars implementing a semi-Dirac cone. If we optically induce a vacancy-like defect in the lattice,…
Three-dimensional topological semimetals can support band crossings along one-dimensional curves in the momentum space (nodal lines or Dirac lines) protected by structural symmetries and topology. We consider rhombohedrally (ABC) stacked honeycomb lattices supporting Dirac lines protected by time-reversal, inversion and spin rotation symmetries. For typical band structure parameters there exists a pair of nodal lines in the momentum space extending through the whole Brillouin zone in the stacking direction. We show that these Dirac lines are topologically distinct from the usual Dirac lines wh…
Moving, merging and annihilating Dirac points are studied theoretically in the tight-binding model on honeycomb lattice with up-to third-nearest-neighbor hoppings. We obtain a rich phase diagram of the topological phase transitions in the parameter space of direction-dependent hoppings. We obtain the conditions for the three Dirac points to merge and for the tricritical points. We find that only very small third-nearest-neighbor hoppings are enough for the existence of the merging of three-Dirac-points and the tricritical points, if the system is sufficiently anisotropic. The density of states…
We consider spin Hall effect in a system of massless Dirac fermions in a graphene lattice. Two types of spin-orbit interaction, pertinent to the graphene lattice, are taken into account - the intrinsic and Rashba terms. Assuming perfect crystal lattice, we calculate the topological contribution to spin Hall conductivity. When both interactions are present, their interplay is shown to lead to some peculiarities in the dependence of spin Hall conductivity on the Fermi level.
We propose a family of free fermion lattice models that have "Dirac loops", closed lines of Dirac nodes in momentum space, on which the density of states vanishes linearly with energy. Those lattices all possess the planar trigonal connectivity present in graphene, but are three dimensional. We show that their highly anisotropic and multiply-connected Fermi surface leads to quantized Hall conductivities in three dimensions for magnetic fields with toroidal geometry. In the presence of spin-orbit coupling, we show that those structures have topological surface states. We discuss the feasibility…
Pseudorelativistic Dirac quasiparticles have emerged in a plethora of artificial graphene systems that mimic the underlying honeycomb symmetry of graphene. However, it is notoriously difficult to manipulate their properties without modifying the lattice structure. Here we theoretically investigate polaritons supported by honeycomb metasurfaces and, despite the trivial nature of the resonant elements, we unveil rich Dirac physics stemming from a non-trivial winding in the light-matter interaction. The metasurfaces simultaneously exhibit two distinct species of massless Dirac polaritons, namely …
The discovery of monolayer graphene has initiated two fertile fields in modern condensed matter physics, Dirac semimetals and atomically-thin layered materials. When these trends meet again in transition metal compounds, which possess spin and orbital degrees of freedom and strong electron correlations, more exotic phenomena are expected to emerge in the cross section of topological states of matter and Mott physics. Here, we show by using ab initio calculations that a monolayer form of transition metal trichalcogenides (TMTs), which has a honeycomb network of transition metal cations, may exh…
Using density functional theory combined with orbital-selective band unfolding techniques, we study the effective band structure of silicene($3\times3$)/Ag(111)($4\times4$) structure. Consistent with the ARPES spectra recently obtained by Feng et al. [Proc. Natl. Acad. Sci. 113, 14656 (2016)], we observe six pairs of Dirac cones near the boundary of the Brillouin zone (BZ) of Ag($1\times1$), while no Dirac cone is observed inside the BZ. Furthermore, we find that these Dirac cones are induced by the interfacial Si-Ag hybridization, mainly composed of Si p$_z$ orbitals and Ag sp bands, which is…
The notion of chiral symmetry for the conventional Dirac cone is generalized to include the tilted Dirac cones, where the generalized chiral operator turns out to be non-hermitian. It is shown that the generalized chiral symmetry generically protects the zero modes (n=0 Landau level) of the Dirac cone even when tilted. The present generalized symmetry is equivalent to the condition that the Dirac Hamiltonian is elliptic as a differential operator, which provides an explicit relevance to the index theorem.
In a recent article [10], the authors proved that the non-relativistic Schrödinger operator with a generic honeycomb lattice potential has conical (Dirac) points in its dispersion surfaces. These conical points occur for quasi-momenta, which are located at the vertices of the Brillouin zone, a regular hexagon. In this paper, we study the time-evolution of wave-packets, which are spectrally concentrated near such conical points. We prove that the large, but finite, time dynamics is governed by the two-dimensional Dirac equations.
The symmetries that protect massless Dirac fermions from a gap opening may become ineffective if the Dirac equation is discretized in space and time, either because of scattering between multiple Dirac cones in the Brillouin zone (fermion doubling) or because of singularities at zone boundaries. Here we introduce an implementation of Dirac fermions on a space-time lattice that removes both obstructions. The quasi-energy band structure has a tangent dispersion with a single Dirac cone that cannot be gapped without breaking both time-reversal and chiral symmetries. We show that this topological …
In monolayer transition metal dichalcogenides, tightly bound excitons have been discovered with a valley pseudospin that can be optically addressed through polarization selection rules. Here, we show that this valley pseudospin is strongly coupled to the exciton center-of-mass motion through electron-hole exchange. This coupling realizes a massless Dirac cone with chirality index I=2 for excitons inside the light cone, i.e. bright excitons. Under moderate strain, the I=2 Dirac cone splits into two degenerate I=1 Dirac cones, and saddle points with a linear Dirac spectrum emerge in the bright e…
We examine the low energy effective theory of phase oscillations in a two-dimensional granular superconducting sheet where the grains are arranged in honeycomb lattice structure. Using the example of graphene we present the evidence for the engineered Dirac nodes in the bosonic excitations: the spectra of the collective bosonic modes cross at the K and K points in the Brillouin zone and form Dirac nodes. We show how two different types of collective phase oscillations are obtained and that they are analogous to the Leggett and the Bogoliubov-Anderson-Gorkov modes in a two-band superconductor. …
Inspired by the great development of graphene, more and more works have been conducted to seek new two-dimensional (2D) materials with Dirac cones. Although 2D Dirac materials possess many novel properties and physics, they are rare compared with the numerous 2D materials. To provide explanation for the rarity of 2D Dirac materials as well as clues in searching for new Dirac systems, here we review the recent theoretical aspects of various 2D Dirac materials, including graphene, silicene, germanene, graphynes, several boron and carbon sheets, transition metal oxides (VO2)n/(TiO2)m and (CrO2)n/…
We consider a two-dimensional honeycomb lattice of metallic nanoparticles, each supporting a localized surface plasmon, and study the quantum properties of the collective plasmons resulting from the near field dipolar interaction between the nanoparticles. We analytically investigate the dispersion, the effective Hamiltonian and the eigenstates of the collective plasmons for an arbitrary orientation of the individual dipole moments. When the polarization points close to the normal to the plane the spectrum presents Dirac cones, similar to those present in the electronic band structure of graph…
The quantum Hall effect in graphene is regarded to be involving half-integer topological numbers associated with the massless Dirac particle, this is usually not apparent due to the doubling of the Dirac cones. Here we theoretically consider two classes of lattice models in which we manipulate the Dirac cones with either (a) two Dirac points that have mutually different energies, or (b) multiple Dirac cones having different Fermi velocities. We have shown, with an explicit calculation of the topological (Chern) number for case (a) and with an adiabatic argument for case (b) that the results ar…
The interband optical absorption of linearly polarised light by two-dimensional (2D) semimetals hosting tilted and anisotropic Dirac cones in the bandstructure is analysed theoretically. Super-critically tilted (type-II) Dirac cones are characterised by an absorption that is highly dependent on the incident photon polarisation and frequency, and is tunable by changing the Fermi level with a back-gate voltage. Type-II Dirac cones exhibit open Fermi surfaces and large regions of the Brillouin zone where the valence and conduction bands sit either above or below the Fermi level. As a consequence,…
A discrete-time Quantum Walk (QW) is essentially an operator driving the evolution of a single particle on the lattice, through local unitaries. Some QWs admit a continuum limit, leading to well-known physics partial differential equations, such as the Dirac equation. We show that these simulation results need not rely on the grid: the Dirac equation in $(2+1)$--dimensions can also be simulated, through local unitaries, on the honeycomb or the triangular lattice. The former is of interest in the study of graphene-like materials. The latter, we argue, opens the door for a generalization of the …
Artificial honeycomb lattices with Dirac cone dispersion provide a macroscopic platform to study the massless Dirac quasiparticles and their novel geometric phases. In this paper, a quadruple-degenerate state is achieved at the center of Brillouin zone (BZ) in a two-dimensional honeycomb lattice phononic crystal, which is a result of accidental degeneracy of two double-degenerate states. In the vicinity of the quadruple-degenerate state, the dispersion relation is linear. Such quadruple degeneracy is analyzed by rigorous representation theory of groups. Using method, a reduced Hamiltonian is o…
While the quantum Hall effect in graphene has been regarded as a realization of the anomaly associated with the massless Dirac particle carrying half the usual topological integer, this is hidden due to the doubling of the Dirac cones. In order to confirm the half-integer contribution from each Dirac cone, here we theoretically consider a lattice model in which the relative energy between the two Dirac points is systematically shifted. With an explicit calculation of the topological (Chern) number, we have demonstrated that each Dirac cone does indeed contribute to the Hall conductivity as the…
We compute the dynamical polarizability of graphene beyond the usual Dirac cone approximation, integrating over the full Brillouin zone. We find deviations at $\hbarω=2t$ ($t$ the hopping parameter) which amount to a logarithmic singularity due to the van Hove singularity and derive an approximate analytical expression. Also at low energies, we find deviations from the results obtained from the Dirac cone approximation which manifest themselves in a peak spitting at arbitrary direction of the incoming wave vector $\q$. Consequences for the plasmon spectrum are discussed.
Recently, a concept of orthogonal metal has been introduced to reinterpret the disordered state of slave-spin representation in the Hubbard model as an exotic gapped metallic state. We have extended this concept to study the slave-spin representation of Hubbard model on the honeycomb lattice.[Phys. Rev. B \textbf{86}, 165134 (2012)] It is found that a novel gapped metallic state coined orthogonal Dirac semimetal is identified. Such state corresponds to the disordered phase of slave-spin and has the same thermal-dynamical and transport properties as Dirac semimetal but its singe-particle excita…
We performed low temperature scanning tunneling microscopy (STM) and spectroscopy (STS) studies on the electronic properties of (R3xR3)R30° phase of silicene on Ag(111) surface. We found the existence of Dirac Fermion chirality through the observation of -1.5 and -1.0 power law decay of quasiparticle interference (QPI) patterns. Moreover, in contrast to the trigonal warping of Dirac cone in graphene, we found that the Dirac cone of silicene is hexagonally warped, which is further confirmed by density functional calculations and explained by the unique superstructure of silicene. Our results de…
We have designed three-dimensional models of topological insulator thin films, showing a tunability of the odd number of Dirac cones on opposite surfaces driven by the atomic-scale geometry at the boundaries. This enables creation of a single Dirac cone at the $Γ$-point as well as possible suppression of quantum tunneling between Dirac states at opposite surfaces (and gap formation), when opposite surfaces are geometrically differentiated. The spin texture of surface states was found to change from a spin-momentum-locking symmetry to a progressive loss in surface spin polarization upon the int…
We theoretically investigate the indirect signatures of the tilted anisotropic Dirac cones on Ruderman-Kittel-Kasuya-Yosida (RKKY) exchange interaction in a two dimensional polymorph of boron atoms. Unlike the case of isotropic non-tilted Dirac material-graphene, here we observe that the tilting of the Dirac cones exhibits a significant impact on the RKKY exchange interaction in terms of the suppression of oscillation frequency. The reason can be attributed to the behavior of the Fermi level and the corresponding density of states with respect to the tilting parameter. The direct measurement o…
We calculate conductance and noise for quantum transport at the nodal point for arbitrarily tilted and anisotropic Dirac or Weyl cones. Tilted and anisotropic dispersions are generic in absence of certain discrete symmetries, such as particle-hole and lattice point group symmetries. Whereas anisotropy affects the conductance g, but leaves the Fano factor F (the ratio of shot noise power and current) unchanged, a tilt affects both g and F. Since F is a universal number in many other situations, this finding is remarkable. We apply our general considerations to specific lattice models of straine…
We theoretically study the effect of the motion and the merging of Dirac cone on the interlayer magnetoresistance in multilayer graphene like systems. This merging, which could be induced by a uniaxial strain, gives rise in monolayer Dirac electron system to a topological transition from a semi-metallic phase to an insulating phase where Dirac points disappear. Based on a universal Hamiltonian proposed to describe the motion and the merging of Dirac points in two dimensional Dirac electron crystals, we calculate the interlayer conductivity of a stack of deformed graphene like layers using Kubo…
Spin-1 Dirac dispersion and Chern insulating phases in 2D honeycomb Sierpiński fractal
Indian Institute of Science Education and Research (IISER) Tirupati（印度科学教育与研究研究所蒂鲁帕提分校）；Kwansei Gakuin University（关西学院大学）；National Institute for Materials Science (NIMS)（国立材料研究所）
We study, theoretically and experimentally, optical properties of different types of honeycomb photonic structures, known also as `photonic graphene'. First, we employ the two-photon polymerization method to fabricate the honeycomb structures. In experiment, we observe a strong diffraction from a finite number of elements, thus providing a unique tool to define the exact number of scattering elements in the structure by a naked eye. Then, we study theoretically the transmission spectra of both honeycomb single layer and 2D structures of parallel dielectric circular rods. When the dielectric co…
The electronic band topology of monolayer $β$-Sb (antimonene) is studied from the flat honeycomb to the equilibrium buckled structure using first-principles calculations and analyzed using a tight-binding model and low energy Hamiltonians. In flat monolayer Sb, the Fermi level occurs near the intersection of two warped Dirac cones, one associated with the $p_z$-orbitals, and one with the $\{p_x,p_y\}$-orbitals. The differently oriented threefold warping of these two cones leads to an unusually shaped nodal line, which leads to anisotropic in-plane transport properties and goniopolarity. A slig…
The study of the magnonic thermal Hall effect in magnets with Dzyaloshinskii-Moriya interaction (DMI) has recently drawn attention because of the underlying topology. Topological phase transitions may arise when there exist two or more distinct topological phases, and they are often revealed by a gap-closing phenomenon. In this work, we consider the magnons in honeycomb ferromagnets described by a Heisenberg Hamiltonian containing both an out-of-plane DMI and a Zeeman interaction. We demonstrate that the magnonic system exhibits temperature (or magnetic field) driven topological phase transiti…
A novel gapped metallic state coined orthogonal Dirac semimetal is proposed in the honeycomb lattice in terms of $Z_{2}$ slave-spin representation of Hubbard model. This state corresponds to the disordered phase of slave-spin and has the same thermaldynamical and transport properties as usual Dirac semimetal but its singe-particle excitation is gapped and has nontrivial topological order due to the $Z_{2}$ gauge structure. The quantum phase transition from this orthogonal Dirac semimetal to usual Dirac semimetal is described by a mean-field decoupling with complementary fluctuation analysis an…
We review recent work of the authors on the non-relativistic Schrödinger equation with a honeycomb lattice potential, $V$. In particular, we summarize results on (i) the existence of Dirac points, conical singularities in dispersion surfaces of $H_V=-Δ+V$ and (ii) the two-dimensional Dirac equations, as a large, but finite time, effective description of $e^{-iH_Vt}ψ_0$, for data $ψ_0$, which is spectrally localized at a Dirac point. We conclude with a formal derivation and discussion of the effective large time evolution for the nonlinear Schrödinger - Gross Pitaevskii equation for small ampli…
We theoretically predict that a large variety of Dirac-cone materials can be constructed in Bi${}_{1-x}$Sb${}_{x}$ thin films, and we here show how to construct single-, bi- and tri- Dirac-cone materials with various amounts of wave vector anisotropy. These different types of Dirac cones can be of special interest to electronic devices design, quantum electrodynamics and other fields.
Muffin-tin methods have been instrumental in the design of honeycomb lattices that show, in contrast to graphene, separated s and in-plane p bands, a p orbital Dirac cone, and a p orbital flat band. Recently, such lattices have been experimentally realized using the 2D electron gas on Cu(111). A possible next avenue is the introduction of spin-orbit coupling to these systems. Intrinsic spin-orbit coupling is believed to open topological gaps, and create a topological flat band. Although Rashba coupling is straightforwardly incorporated in the muffin-tin approximation, intrinsic spin-orbit coup…
Combining first-principles calculations and spin- and angle-resolved photoemission spectroscopy measurements, we identify the helical spin textures for three different Dirac cone states in the interfaced systems of a 2D topological insulator (TI) of Bi(111) bilayer and a 3D TI Bi2Se3 or Bi2Te3. The spin texture is found to be the same for the intrinsic Dirac cone of Bi2Se3 or Bi2Te3 surface state, the extrinsic Dirac cone of Bi bilayer state induced by Rashba effect, and the hybridized Dirac cone between the former two states. Further orbit- and atom-resolved analysis shows that s and pz orbit…
